Solving Dense Nonsymmetric Linear Systems Arising from Integral Formulations - Magnetics, IEEE Transactions on

Integral formulations inhererikly lead to dense systems of equations. For this reason, integral formulations are oftfen thought t,o be computtationally t,oo expensive t,o be considered as a practical approach. To make integral met,hods more appealing effort. must, be devoted to solving large dense linear systems efficiently. There are several factors to be taken into account, when considering a solver: (1) t,he number of operations it needs, including it’s 0-complexity, (2) its storage requirenierits and ( 3 ) its implementation in the comput,er. I n t,his paper we consider these factfors for several solvers. We present results using three protilems: TEAM problems 13 and 20 and a dipole magnet. The labels T13,,T20,, and DIP, will be used for referring, to 71 x 7% matrices arising from these problems. All the rmults were computed on a DEC 3000-700 AXP workstation. Our goal in this paper is tc, discuss various issues we have encountered in trying to find and implement solvers in the GFUNET package [l]. GFUNET is based on a n h-type volume integral formulation. The A matrices produced by GFUNET are nons;ymmetric. Section I1 describes shortly the BLAS library wliidi contains routines for efficiently computing matrix and vector operations. One class of iterative solvers for linear systems is the Krylov class. We discuss these in Section 111. The performance of iterative methods caii often be dramatically improved if the user can find an appropriate preronditioner. This is the topic of Sectmion IV. In section V we discuss the wavelet transformation. We tested this transformation to see if i t would “sparsify” our original dense matrix. In the final two sections we present test results and conclusions.